182 ■ On the Effect of Internal Friction on Resonance, 



If 



. nl _ ~ . ri*lk 



sin -= 0) Q=±^, 



A^O, and B 1= = H — ^ A, 



r A noc 



cb = + -r x cos — , 



1 , , 2«A . nx 

 I -ur= H — — - sin — • 

 [_ r - n 2 /A: a 



A 2 

 Work done = -^fE£. 



We infer that the energy imparted to the string varies as the 

 square of the amplitude of vibration of the extremity, that it 

 rapidly increases as the period approaches that of the string, 

 that, if these periods differ materially, the work is directly pro- 

 portional to the friction and increases rapidly with the number 

 of vibrations — but that if the periods are identical, the work 

 varies inversely as the friction, the diminishing of the friction 

 being more than counterbalanced by the increased amplitude. 



It is interesting to examine how this energy is distributed 

 over the string. This is easily done by writing down the work 

 done by one portion of the string from x to I, on the remainder 

 from to a, and then taking the differential ; we readily find that 

 work absorbed by portion dx of string 



=^'(2l'tfl> 



Substituting, we obtain, when the string does not resonate, 



2 nx 

 cos' 1 — 



, n 4 k~Et 

 work=— jh. z dx) 



2a . 9 nl 



snr — 



when the string resonate 



c 



= §cos*-.AV*. 

 rk a 



In either case the absorption of energy, and therefore the 

 heating- effect, is greatest at the nodes, and, omitting squares of 

 k, vanishes at the middle of the ventral segments. Directly the 

 contrary will result from the friction of the string against the air. 



Glass Works, near Birmingham. 



